Cyclic sieving phenomenon on dominant maximal weights over affine Kac-Moody algebras

We construct a (bi)cyclic sieving phenomenon on the union of dominant maximal weights for level ℓ highest weight modules over an affine Kac-Moody algebra with exactly one highest weight being taken for each equivalence class, in a way not depending on types, ranks and levels. In order to do that, we introduce S-evaluation on the set of dominant maximal weights for each highest modules, and generalize Sagan's action in [17] by considering the datum on each affine Kac-Moody algebra. As consequences, we obtain closed and recursive formulae for cardinality of the number of dominant maximal weights for every highest weight module and observe level-rank duality on the cardinalities.